Exact relativistic theory of wave propagation in prestressed nonlinear elastic solids

A NNALES DE L ’I. H. P., SECTION A

G ÉRARD A. M AUGIN

Exact relativistic theory of wave propagation in prestressed nonlinear elastic solids

Annales de l’I. H. P., section A, tome 28, n

^o

2 (1978), p. 155-185

<http://www.numdam.org/item?id=AIHPA_1978__28_2_155_0>

© Gauthier-Villars, 1978, tous droits réservés.

L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.

org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

155

Exact relativistic theory

of

wave

propagation

in prestressed ^nonlinear elastic solids

Gérard A. MAUGIN

Universite de Paris VI,

Laboratoire de Mecanique Theorique ^{associe au C. N. R. S.,}

Tour 66, 4, Place Jussieu, ⁷⁵²³⁰ Paris, ^Cedex 05, ^France

Vol. XXVIII, ^n° 2, 1978, Physique ^’ théorique. ^’

ABSTRACT. ^- The

propagation

of weak discontinuities in

general

relativistic

prestressed

elastic solids is studied under the sole

hypothesis

that the material be

isotropic

ⁱⁿ

^an ideally

unstrained state. No limitations

are

placed

upon the

amplitude

^{of stresses and} deformation _processes and the formulation has

thermodynamical

foundations. In this

general

^frame-

work where the state

equation

^{for the}

potential

^{has a}

quite arbitrary form,

it is shown that

(i) principal

^{wave fronts are either}

longitudinal

^{or transverse}

(the propagation

^of

longitudinal

^ones

being impeded by

^the

possible incompressibility

^{of the}

material); (ii)

ⁱⁿ

general,

^there may be two transverse waves with distinct

speeds; (iii)

the values of these

speeds

^is

expressible

in terms of three

(scalar)

response functions

(typical

^{of the}

material)

^and

of the initial

stretches; (iv)

^{it is}

possible

^to

give

^a relative classification of these values and

(v)

^{in the case of}

propagation through

^{an initial state of}

high hydrostatic

pressure

(case

of dense stellar

objects),

^{there exists a}

universal

relationship

^{betw een the}

simple speed

^of

longitudinal

^distur-

bances and the double

speed

^of transverse ones and the

speed

^{of sound of a}

relativistic

perfect

fluid that would have a law of

compression corresponding

to the initial state. For a sensible

special

^{case of state}

equation

^{and for an}

initial state of

hydrostatic

pressure, the

speeds

^of

propagation

^{and the}

speed

of sound referred to above are determined

exactly

in function of two

fundamental scalars of the material and a

density

^ratio.

Taking

^account

in

supplement

^of

perturbations

^{in the} geometry ^of

space-time,

^{the same}

formalism is

applied

^{to the} construction of

perturbation equations

l’Institut Henri Poincaré - Section A - Vol. XXVIII, ^{n° 2 - 1978.}

(generalized

Hooke-Navier-Duhamel

equations)

^valid either for isothermal

or

isentropic

processes. The latter

equations

^{are those to be} used either in the

study

^{of « elastic »}

gravitational

^{wave detectors or in the}

study

^of

small elastic oscillations of dense stellar

objects.

RESUME. 2014 La

propagation

des discontinuites faibles dans les solides

elastiques precontraints

^{est etudiee} dans Ie cadre de la relativité

generate

sous la seule

hypothèse

que Ie materiau soit

isotrope

^{dans un etat ideal}

non deforme. Aucune restriction n’est

imposee

^a

1’amplitude

^{des deforma-}

tions et des contraintes et la formulation est basee sur la

thermodynamique.

Dans ce cadre

general

^ou

1’equation

^{d’etat a une}

expression

suffisamment

arbitraire,

^{il est} montre que :

(i)

^{les ondes}

principales

^{sont soit}

longitudi-

nales soit transversales

(Fincomprcssibilite possible

^{du milieu}

empechant

la

propagation

^des

premières); (ii)

^il y ^{a en}

general

^{deux ondes} transversales de vitesses

distinctes ; (iii)

^{la valeur de ces vitesses} peut ^etre

exprimee

^en

fonction de trois fonctions de

reponse

^scalaires

(typiques

^du

materiau)

^et

des

elongations initiales ; (iv)

^{il est}

possible

^{de donner une} classification relative de ces vitesses et,

(iv)

^{dans Ie cas ou la}

propagation

^{a lieu dans un}

etat de forte

pression hydrostatique (cas

^des

objets

stellaires

denses),

^il

existe une relation universelle entre la vitesse

simple

^des

perturbations longitudinales,

la vitesse double des

perturbations

transversales et la vitesse du son d’un fluide

parfait

relativiste

qui

^{aurait une loi de}

compression correspondant

^{a l’état initial. Pour une}

expression plausible

^de

1’equation

d’etat et un etat initial de

pression hydrostatique,

^{il est alors}

possible

^de

determiner exactement les vitesses de

propagation

^ainsi que ^cette vitesse

sonique

^en fonction de deux scalaires

caracteristiques

du materiau et d’un rapport de densites. De

plus,

prenant ^en compte ^les

perturbations

^de

la

geometrie

^de

l’espace-temps,

^{Ie meme} formalisme est

employe

^{a la}

construction des

equations

^de

perturbation (equations generalisees

^de

Hooke-Navier-Duhamel)

^valables pour des processus isothermes ou

isentropiques.

^Les

equations

^{ainsi obtenues sont celles}

qui

^{doivent etre}

utilisees soit dans l’étude des vibrations

elastiques

des detecteurs d’ondes

gravitationnelles,

soit dans l’étude des

petites

oscillations

elastiques

^des

objets

stellaires denses.

1. INTRODUCTION

In papers

[1 ]-[2] recently published

^{we have studied the}

propagation

of infinitesimal discontinuities in certain

simple

^{classes of} relativistic elastic

(or hypoelastic)

^{bodies. A}

^study

^{of the}

growth

^{of the}

amplitude

^of

such wave fronts

along

^{their ray} has demonstrated the need for ^an exact

theory requiring

^no

hypotheses

^{as far as the}

amplitude

of strains is concerned and

relying

upon ^a sound

thermodynamics.

It is the purpose of _{this paper}

de Henri Poincaré - Section A

to present ^{such an exact}

theory

with the sole

hypothesis that,

^{in the case}

of

arbitrary

^finite

strains,

^the

body

^be

isotropic

^{in an}

ideally

^unstrained

state. The wave

fronts, however,

propagate

through

^an

initially

^strained

state. For

analytical

convenience the

study

^must be limited to that concern-

ing

^a

propagation along

^a

principal

direction of the initial state of stress.

Since the state

equations

of the elastic bodies to which the present ^treatment

applies

are, for the

least, badly

^known

(e.

g., in neutron

stars),

^{we concen-}

trate upon the derivation of those results which _may be said ^to be universal in the sense that

they

^{do not}

depend explicitly

^{on the exact form of such a}

state

equation.

Having

^{recalled some} basic notions of the

theory

of deformation processes for

general

relativistic matter in

space-time

^{in Section}

2,

^we establish in Section 3 the exact form of constitutive

equations

for relati- vistic

isotropic

nonlinear thermoelastic solids.

Following

^{the definition}

of infinitesimal discontinuities in Section

4,

^we prove in Section 5 a series of

lemmas,

theorems and corollaries

concerning

^universal

(in

^{the sense}

specified above)

^results

pertaining

^{to the}

longitudinal

^or transverse character of the wave

fronts,

the values of the

speed

^{of these waves, and} the relative classification of these values. In

general

two transverse waves with distinct

speeds

^{and one}

longitudinal

^{wave can} propagate. ^In

particular,

^{a universal}

relationship

between the

propagation speeds

^of

longitudinal

^{and transverse}

wave fronts in an initial state of

high hydrostatic

pressure

(as

may ^occur in certain

astrophysical objects)

^is proven in this

general

framework. The

case of relativistic

incompressible

nonlinear elastic solids which

requires special

attention is

briefly

commented upon in Section 6 where the non-

propagation

^of

longitudinal

^wave fronts is _proven for such bodies. The results are

specialized

^{in Section 7 for a}

special

^{form of the free} energy

density.

^{There the}

speeds

^of

propagation

^of

longitudinal

^{and transverse}

wave fronts and the

speed

of sound in the case of an initial state of

hydro-

static _pressure are determined

exactly

^{in terms of two} fundamental scalars

(analogous

^{to Lame’s}

moduli),

^{which are} characteristic of the

material,

and of the

density

^ratio.

By

way of

conclusion,

^we deduce in Section

8,

from the exact

expressions

established

before,

^the

generalization

^{of Hooke’s}

law to be used in the treatment of small elastic oscillations either in

« elastic »

gravitational-wave

^{detectors or in}

astrophysical objects

^acted

upon

by

^{their own}

gravitational

field. The

Appendices provide

^{lists of}

coefficients as also a brief

comparison

between constitutive

equations proposed by

^{different authors for the}

description

of relativistic elastic

matter

( 1 ).

Basic results

reported

in this paper have been enunciated in a

short Note

[4].

^Related

previous

^works

using

^{a different}

formalism,

^which

e) ^The brief review given ^{in a} preceding paper ([3], Appendix) ^written in 1971-1972

now is obsolete.

Vol. XXVIII, n° 2 - 1978.

either are based on too

specialized hypotheses

^or fall short of the conclusions reached in the present paper, ^are those of Bressan

[5],

^Grot

[6] (in special relativity),

^{and Carter}

[7].

2.

PREREQUISITES

2.1. Notation

Let M ⁼

(V4,

^{be a}

space-time

^of

general relativity equipped

^with

a normal

hyperbolic metric

ga~

(x, ~ == 1, 2, 3, 4 ;

^{index 4}

time-like ;

Lorentzian

signature + , + , + , - ).

^{ux is the}

four-velocity

^{such that}

1

(c

^{= 1 for} notational

convenience). ~~

^and

oa

denote the

partial

^and covariant derivatives with respect ^{to the local} chart x03B1 of M.

D ⁼

u03B1~03B1

is the invariant derivative in the direction of M".

P03B103B2

⁼ g03B103B2 ⁺ u03B1u03B2 is the

spatial projector

which is used

systematically

^{in the}

following

^deve-

lopment

^{to write down} the local canonical

space-time decomposition

^of

any ^tensor field defined on M. The local

spatial projection

^of any

geometrical object

^{A is noted}

A1

^{and admits u as zero vector} for all its indices in a local chart.

Objects

^{such that A =}

A1

^{are said to be}

spatial.

^The transverse or

spatial

covariant derivative is defined

by oa

⁼

22 Deformation of matter in

space-time

Following Maugin [8]

and Carter and

Quintana [9],

^we admit that the motion of a relativistic continuum is described either

by

^{means of a canonical}

differentiable

projection P

such that P :

G[B] ~

^{N or} with the aid of the

space-time parametrized

congruence of world lines ~ : ^{x =}

~(X, T),

X E

B,

^{1" e !R. Here}

F[B]

^is the open tube of y4 which is swept ^out

by

^the

material

body

^B

(whose

constituents are the material ^«

particles » X)

and

(V3, K,

^{L =}

1, 2, 3,

^{is the} three-dimensional manifold which serves to describe the material continuum. B is an open

region

of T is the _proper time of X. ~ is

equipped

^{with the local}

background

metric

GKL

and local charts

XK,

^{K =}

1, 2,

3. We have thus

These relations ^are assumed to possess ^a sufficient

degree

^of

continuity

and

differentiability

^{in their} arguments ^{so as to allow for the}

forthcoming manipulations.

^For

instance,

^{one can define} the inverse motion

gradient X~

by

°‘~ being

^the

reciprocal

^{of a}

space-time

invariant noted

C L

^but

tensor field on ~~~ 2014 is constructed

by applying

^the

projection ~ :

Annales de l’Institut Henri Poincaré - Section A

This defines the relativistic

analogue

of the Piola finite-strain tensor of classical continuum mechanics

(Compare [lO], Chap. I).

^Its

geometrical significance

is clear. It is the

image

^{of the}

space-time

^metric

by

^the

projection

^{of the}

space-time

^{on its}

quotient by

the congruence

(2.1)3’

Furthermore, assuming

^{that the} Jacobian determinant of

(2 .1 )3 keeps

the same

sign (e.

g.,

plus)

^{in the course} of the relativistic motion of X and

defining

the direct motion

gradient xK by

the chain rule of differentiation

yields

and it is

possible

^to define the strain tensor

CKL

^{such that =}

ði

and

Let

Gxa

^{be the}

image

^of

GKL

^{in M}

by f!JJ, i.

e.,

Then we can define two useful tensor fields which serve to measure finite strains either on N or on M,

EKL

^{and such that}

[3] [77] ]

and

Elementary computations [3], [72]

^{then allow one to} establish the

following

results that relate

propertime

^{rates of}

change

of different

geometrical objects

^{of interest :}

and where

and

Vol. XXVIII, ^n° 2 - 1978. ₁₁

for _any

spatial

^tensor

Aaa

^{= denotes the} Lie derivative with respect ^{to the field u. In terms of the} differentiable

projection 9,

^{we have}

Equation (2 .12)2

^{is a} consequence of

Eqs. (2.12)i, (2 . 7)

^and

(2.10).

^This

shows that

G«~

^{is the}

background

^{metric on}

M,

^which serves as a local standard to measure strains.

According

^to

Eqs. (2.12)

^and

(2.14),

^the

Herglotz-Born

local condition of

rigid-body

motion is defined in diffe- rential form

(Killing’s theorem) by

^{either one of the}

following

conditions :

2.3. Field

equations

In

supplement

^to Einstein’s field

equations

that relate

linearly

^the

Einstein tensor and the total energy-momentum tensor, ^we have

Here

p(x)

⁼

p(X, T)

^{is the mass} per unit of _proper volume. In absence of heat

conduction, electromagnetic

fields and

spin

^the energy-momentum

tensor admits the

following simple

^canonical space-time decomposi- tion :

where t03B103B2 = t03B203B1, as a consequence of

Eq. (2.20)2,

^{is the}

spatial

relativistic

stress tensor, ^{and E} is the internal energy per unit _{of proper} mass.

Taking

account of

Eq. (2.19)

^{and of the fact that =}

0,

^we

project Eq. (2.20)i along

u~ and

orthogonally

^{to u to obtain}

and where

is the tensorial index of the

continuum,

^{cf. Ref.}

[2].

^{In the} present ^{case the} local statement of the second

principle

^of

thermodynamics

^{reduces to the}

equation Dry

⁼

0, where ~

^{is the} entropy per unit of proper ^mass.

Introducing

the

specific

^free energy

t/J by

^{= s 2014 where 8 is} the proper thermo-

dynamical

temperature

(o

&#x3E;

0,

^{inf 8 =}

0),

^{we can rewrite}

Eq. (2.22)

ⁱⁿ

the form

Annales de l’Institut Henri Poincaré - Section A

3. CONSTITUTIVE

EQUATIONS

FOR ISOTROPIC THERMOELASTIC BODIES

A natural definition for

general

thermoelastic bodies

(i.

e., with a

priori large deformations)

^is

given by postulating

^a functional

dependence

^{of the}

form

.

That

is,

^{there are no}

hereditary

effects since

dependent

^and

independent

variables are considered at the same event

point

^{of M} or,

equivalently,

for the same values of the four parameters

(XK, -r).

^{Then we have the}

THEOREM 3 .1. ^- The exact constitutive

equations o, f ’

^an

anisotropic

thermoelastic

body

^are

given by

03C8 satisfying

^{0 the}

following

^{0 set}

of first

^{order linear}

partial differential

tions:

It then follows the

L __

COROLLARY 3.2.

Equations (3.2)

^and

(3.3)

^{can be}

replaced by

i. _e., the image by P

of TKL ~ (~03C8/~ Ch) = TLK

up to the

2014

Consider ~

⁼

0)

^{to start}

^with,

^then compute ^with the

help

^of

Eq. (2.10)i

^{to obtain}

if c~ =

Furthermore, according

^{to the}

principles

of formulation

set forth

by

^the

Author, 03C8

^{must be}

objeetive,

^i. e., its

explicit

form should

not

depend

^on the observer

[12], [l3].

^{We have shown that in the} present

case which does not involve

hereditary effects,

this invariance is

equivalent

to the rotational Lorentz invariance of

L~

^{in a} local inertial

frame,

^or

else,

^to invariance under all generators ^of

SO(3)

ⁱⁿ local nonholonomic

spatial

^frames

along

^~.

Studying

^{such a form} invariance for

8)

^under

infinitesimal

transformations xa ⁼⁼⁼

^{+ in}

special relativity

^or

under infinitesimal rotations ^{= =}

1, 2, 3,

^between

rigid

Vol. XXVIII, n° 2 - 1978.

spatial

^{triads at an event}

point

^of

M,

^{where * indicates}

validity

ⁱⁿ

inertial frames

only,

^{e is an}

infinitesimally

small and

L~a

^is

arbitrary,

^and

recasting

^the

resulting equation

^{in a}

complete

^covariant

framework,

^{we are led to the condition}

(3 . 3).

^{On account of}

this, Eq. (3.5) simplifies

^{and the}

expression

^of

03C1D03C8 being

^{carried in}

Eq. (2.25)

^{which is}

posited

^to be valid for _any DB and all deformation fields that do not

rigidify

the continuum in the sense of

Herglotz

^{and Born}

(i.

^e.,

0), completes

the

proof

of Theorem 3.1. The system of differential

equations (3.3) integrates immediately along

^its characteristics if

~ depends

^on

X~ only through

^the

space-time

^invariant combination Hence the

proof

^of

Corollary ^3.2; Q.

^{E. D.}

Equations (3.4)

^{are the}

equations

^deduced pre-

viously

^{from a} variational

principle by

the Author

[8].

Equations (3.4)

^{and those}

equivalent equations

^{which are discussed}

in

Appendix

^{I describe}

anisotropic

thermoelastic bodies. The notion of material symmetry which relies _upon

cristallography

^is

essentially

^{a three-}

dimensional Euclidean

notion,

^i. e., it concerns the

study

^{of the invariance}

of functions with respect ^{to members of}

subgroups

^of the group

0(3).

Since,

^{as a} result of

Corollary 3.2, ~r depends

^now

only

^on arguments defined on 8

being

^a parameter, ^material symmetry ^must be discussed in the local tangent space ^to ~l at X.

However,

^we shall avoid this

compli-

cation in the

sequel

^{for we shall use}

only

arguments ^{defined on M to faci-} litate the

analysis

of wave-front

propagation.

^In

^fact, using

^{the result}

enunciated in Theorem

3.1,

^{we can state the}

THEOREM 3.3. ^-

a)

^{The exact} constitutive

equations of’

^an

isotropic

relativistic thermoelastic

body

^are

given by

^either

or

depending

^on

whether ~

^{or E is used as}

thermodynamical potential.

b)

^{and B are}

isotropic functions [in

^{the sense ’ of}

SO(3)]

of

^the

relativistic finite-strain

^{tensor ~.}

Proof

^{2014 We}

can write Eq. (2.9)

in the form

Note that

depends

^on

X~

^via

X~

itself and via

CMN

^that

depends

^on

~

hence on its

reciprocal X~ .

^{It follows}

by varying Eq. (2.5)i

1 that

Anna/es de Henri Poincaré - Section A

From

Eq. (2.6)

^it follows that

Then

Eq. (3.8) yields

on account of

Eqs. (2.5). ^Then,

^{if one makes the}

change

^of

independent

variables

~(X~0) ~ Eq. (3.11)

substituted in

Eqs. (3.2)i

and

(3 . 3) yields Eq. (3.6)1 and,

^{on account of the} symmetry ^{of the}

following

system ^{of first} order linear

partial

differential

equations

similar

equations involving 8

^instead

of 03C8

^{are obtained}

by performing

the

partial Legendre

transformation e

= ~

^{+ As is}

readily checked, Eq. (3.12)

^{is but the covariant}

expression

^{of the fact}

that ~

^{must be}

objective,

i. _e., form-invariant

by SO(3)

^{in a local inertial} frame. In such a frame

Eq. (3.12)

is satisfied

identically if 03C8 depends

^{on E}

only through

^{its funda-}

mental invariants

Ik

^{= tr}

~k,

^{tr =}

^{trace, k}

⁼

1, 2,

^{3. Since these are} space- time

invariants,

^the result holds

good

ⁱⁿ

fully

covariant formalism. This

means

that 03C8

^{or E is an}

isotropic

function of its tensorial argument, 8 or ~

acting

^{as a}

simple

parameter. ^The

body

^thus described exhibits ^no

preferred spatial

^{direction as far as its} response ^to deformations is concerned. It is

isotropic ; Q.

^{E. D.}

Applying

^the

Cayley-Hamilton

theorem it then is

possible

to restate the

foregoing

^{result as}

COROLLARY 3.4. ^- The ^exact constitutive

equations of

^an

isotropic

relativistic thermoelastic

body

^are

given in

intrinsic notation

by

where the

gr’s

^are

space-time

^{invariant scalars}

and, by

convention,

(~°)°‘a =

On account of the

expression given

ⁱⁿ

Appendix

II for the scalars _gr,

Eq. (3.13)1

^{is the} relativistic version of the constitutive

equation

^derived

by Murnaghan [l4]

in classical

isotropic elasticity

with finite deformations.

Remark that this

equation

^{is universal in the sense that 8 is a}

general

^func-

tion of the invariants

Ik,

^whose

expression

^{can be} constrained

only by

^some

regularity assumptions,

^some conditions of elastic

stability

^{and the condi-}

tions of relativistic

causality

^{and the}

required reality

^{of wave}

speeds,

^the

latter

being

determined in

following

^sections.

Vol. XXVIII, ^n° 2 - 1978.

Remark.

2014 (f)

^{The manner} in which the

equations

above have been obtained guarantees that

they

^{are valid in}

_special relativity,

^{and at the}

nonrelativistic

limit,

in classical continuum

mechanics,

^and

that,

ⁱⁿ

supple-

ment to the

objectivity requirement

^{of the}

Author, they satisfy identically

the

rheological

invariance

proposed by Oldroyd [15]

^for

general

relativistic continuous matter.

Remark.

- (ii)

^{A direct}

proof

^of

Eqs. (3.13)

^{can be}

given by starting

with the a

priori

functional

dependence ~(~ap, 8). Then,

ⁱⁿ

computing

one uses

Eq. (2.16)

^to pass from to hence to

daa

^{in virtue}

of

Eq. (2.14). Taking

^{account of the}

decomposition

^of ea~ ⁱⁿ

symmetric

and

skewsymmetric

parts, ^one is thus led to

Applying

^{the same} argument ^{as that}

applied

^{in the}

proof

of Theorem

3 .1,

but for the

objectivity of 03C8

^{as a function of it results}

_Eq. (3.12).

^Hence

Eq. (3.14) simplifies,

^{and it remains to} substitute for the

expression

^of

provided by

^this

simplified equation,

^into

Eq. (2.25)

^{to arrive at the}

results

(3.6).

4. DEFINITION

OF INFINITESIMAL DISCONTINUITIES

We recall the definitions introduced in a

previous

^work

[16] (See

^also

Lichnerowicz

[77]).

^{Let = 0} be the time-like

hypersurface

^that represents ^a

discontinuity

front which propagates ⁱⁿ V4 and thus separates

~ _

F[B]

^{in two}

subregions

^{and at} each time. We set

and

~ is the

(nondimensional) speed

^{of the}

discontinuity

front measured relati-

vely

^{to the}

moving matter. l"

is oriented from the « minus » to the «

plus »

side of W. A + and A -

being

the uniform limits of A in

approaching

^{W on}

its two

faces, we note [A] =

^{A+ - A-. If}

^A,

g03B103B2 ^{and u03B1 are continuous}

across Wand

if 03B4

denotes the Dirac distribution with compact support

on

W,

^{then we can write} -

and

where ^’ the field ðA is called the infinitesimal

discontinuity

^{of A}

through

^W.

We call the

two-plane

^’

orthogonal

^{to the unit}

spatial

^vector

,

Annales de l’Institut Henri Poincare - Section A

S«~ = P«~ - !~«~’~,a

is the covariant

projector

^{on to}

H~~.

The canonical

decomposition

^of any

spatial geometrical object along

the direction of ~, and on to

H~

^{is obtained}

by applying

^the operator

S,

^e. g., with ^an obvious notation and obvious

properties

for the elements of

decomposition

^thus

introduced,

Similar

decompositions hold ^good

^for

^f«~

^{and with the} elements of

decomposition F«, F)

^and

E«, E), respectively.

We call

Ø’[B]

^c

M) == {p,

^{~, 17, u«, a solution of the}

system ^of

equations

^formed

by

Einstein’s field

equations, Eqs. (2.19)

and

(2.23),

the constitutive

equations (3 .13)

and the condition

D17

^{= 0}

(provided

^{that such a solution}

exists;

this difficult

problem

of existence is not

approached

^{in this}

paper).

^{Then the wave} fronts that we consider in the

forthcoming

^sections

satisfy

^the

following

^{set of}

hypotheses :

Hi :

any

typical

^solution is continuous across

W;

H2 :

except ^{for the metric all}

space-time

derivatives of the first order of the fields of the solution suffer discontinuities across W

(the

case where 0

requires

^a

special study);

H3 :

^{W is not a}

gravitational

^wave

front,

^i. e., ~2 - 1 is

excluded;

.

H4 :

^{W is not a material wave front} or, in other

words,

^since

D17 = 0 yields ~~~

^{= 0 in} agreement ^with

Eq. (4.5)~,

^{W is not an} entropy

front,

i. _e., ~ ⁼ 0 is excluded so that

~r~

^{= 0}

necessarily.

In virtue of

H..1,

^{W is not a shock wave since}

~u«~ ~

^{0. In} virtue of

H3

and

H4

the admissible range for GlC is limited to the _open interval

]0,1[ c= ~

if ~ is to be real and less than the

light velocity

^{in vacuum}

(relativistic causality).

We call

principal

^wave fronts those wave fronts for

which ~,«

^coincides

with an

eigenvector

of the initial state of stress E

According

^to

Eq. (3.13)i,

^{if W is such a wave}

front,

^{then the}

corresponding ~,«

^coincides

also with an

eigenvector

^of the initial state of strain E

Naturally,

this holds true

only

^for

isotropic

^bodies.

Longitudinal

^{wave fronts are those}

wave fronts for which

(~u ~ 0, ~u1

⁼

0),

^and transverse wave fronts are

those for which

(5M

⁼

^0, 0).

^{We shall not consider}

^general

^wave

fronts which _may be called mixed wave fronts

(Cf. [2] ).

5. PRINCIPAL WAVE FRONTS

IN ISOTROPIC RELATIVISTIC THERMOELASTIC BODIES We consider

only principal

^{wave fronts} except ⁱⁿ

degenerate

^{cases of}

initial state of stress where the character or

principalness

^{has no}

meaning.

In

general

admits three distinct

orthogonal (with

respect ^to the metric

eigenvectors (spatial

^unit

four-vectors) ~=1,2,3,

Vol. XXVIII, ^{n° 2 - 1978.}

with

corresponding eigenvalues _~.

^{For a}

principal

^{wave front}

W,

^let

d~ 1,

coincides with A.

d~ 1 ~

^{is also an}

eigenvector

^{of Let}

_{t{ 1 ~}

be the

corresponding

^stress

eigenvalue and E~

^the

corresponding

^strain

eigenvalue. Then t~

^and

E~

^{are related}

by

^the

equation

The

remaining

^two

eigenvectors

^{of both}

_ta~

^{and and}

d~3~,

^{form an}

orthonormal

dyad

^{on which can be}

projected

any tensorial

object

^{A such}

that

S(A)

^{= A. Then we can set the}

following

^lemma.

LEMMA 5.1. ^-

Principal (infinitesimal discontinuity)

^wave

fronts

^W

that

propagate in

^an

isotropic

relativistic thermoelastic

body

^{are either}

purely longitudinal

^or

purely

transversal.

Proof.

^A

straightforward

calculation

yields

^the

following expression (written

^{in intrinsic}

formalism)

^{for the}

right-hand

^{side of}

Eq. (2.23)

^on

account of

Eq. (3.13)1 :

where

gr(Ik, ~) -

^{and the nine scalars which are functions}

of

Ik and ~ only,

^{are listed in}

Appendix

^III.

Now consider the infinitesimal discontinuities of

Eqs. (2.19), (2.23)

and of

(~~

^{on account} of the definition

(2.16)

^{and of}

Eq. (5.2). Taking

account of the fact that

fa~

^{and are} continuous across Wand

using

^the

definitions

(4.1) through (4.5)

^and

Eq. (5.1),

^we

obtain,

with ~ ~

0,

and

Annales de l’Institut Henri Poincaré - Section A

where

n writmg /) ^{we nave} taken accoum 01 me results (5.5), (5. 5) ^anu

(5.6)

^{and of} the fact that

51]

^{= 0.}

Upon using

^the

decompositions ^(4.6)

and

(4. 7)

^{and the}

analogous decompositions

^for

fa~

^and

~a~

^and

^accounting

for the fact

that, being

^an

eigenvector

^{of and} it also is ^an

eigen-

vector of the

projection

^of

Eq. (5.7) along

^the direction of A

yields

~’ F - {(1 - 2~)~ - ~

⁺

2~~1)

⁺

2~(1 - 2~)} ^{~~u = 0, (5 . 9)}

with ₂

whereas its

projection

^{onto reads}

the mixed

projection vanishing identically

^for

fa

⁼

Fa

⁼

Ea

⁼

^{0 ;}

^hence

the

proof

^of Lemma 5 .1. That

is,

^{we have}

uncoupling

^between

longitudinal

and transverse wave fronts because

(i)

^{of the}

isotropy

^{of the}

body

^and

(ii)

of the

principalness

^{of the wave front.}

Since

p ~ 0,

^{we can state at once the}

following

^{theorem :}

THEOREM 5.2. ^-

Longitudinal principal

^wave

fronts

^that

propagate in

an

isotropic

relativistic thermoelastic

body

^{have a}

speed

^{such that}

where

is the

principal

^{stretch in the}

spatial

^{direction whereas} transverse

principal

wave

fronts

ⁱⁿ

generat

^{have two distinct}

speeds, uT2

^and

uT3,

^{which are}

so utions

0

^the

equation

where

Equation (5.14)

^{is solved}

immediately

ⁱⁿ the nonholonomic frame

(d(2~,

where

diagonalizes. Setting .f’2

⁼

F.2, 3

⁼

F?3, ~2

⁼

E.2, ~3 = E33,

and i;, ⁼

( 1 - 2~)’~~ ~’

⁼

^{2, 3,}

^{the solutions of}

(5.14)

^are

given by

Vol. XXVIII, ^{n° 2 - 1978.}

For the wave

speeds

^{to be real and less than}

unity,

^the

right-hand

^side

of

Eqs. (5.12)

^and

(5.16)

^{must be} in the interval

]0,

¹

[.

^This

clearly imposes

constraints on the initial state

9Mo

^i. e., ^on rather

complex

combinations of the _response functions gr and the initial strains and

stretches,

and vk.

Apart

from those

constraints,

the results enunciated in the form of

Eqs. (5.12)

^and

(5.16)

^{are universal since}

they

^{do not}

depend

^on any assump- tion as

regards

^the

amplitude

^{of strains}

(e.

^g.,

they

^{are valid}

for finite strains)

and on any

particular

functional

dependence

^{of the internal} energy ^E

(Ik, ri),

which of course possesses ^a sufficient

regularity.

^A

general study

^{of the}

constraints referred to above cannot be

performed

^{under the}

hypothesis

^of

a

general

^{initial state. Neither can} it be achieved a relative classification of the two transverse wave

speeds

^{in an exact manner in such a}

general

^frame-

work. The

approximate following results, however,

^can be established.

Let us define

Ak, k

⁼

^{2, 3,} by

Then with the definition

of vk

^{we can}

give

^the

following

^{form to the diffe-}

rence

~T2 - ~3:

Of _course,

f2 ~ /3 ==

^{1 +}

0(c-2)

&#x3E;

0,

^{so that we can} introduce a mean

value f

^for

f2 and f3

^{and rewrite}

(5.18)

^as

The

Ak

^{are all}

positive

from their _very

definition,

^and vi &#x3E; v~

yields Ai A~.

We have thus

COROLLARY 5.3.

- a)

Transverse wave

fronts

^with

amplitude paraltet

to the axis

of

^lesser transverse stretch travel at a greater absotute

speed

than others

if,

^with gl 1 &#x3E;

0,

i)

^either g2 0 and 1 -

f(A2

⁺

A3) g1/g21, ii)

^Or g2 &#x3E; 0 and

f(A2

⁺

A3) -

¹

b)

Transverse ^wave

fronts

^with

amptitude parattel

^{to the axis}

of

greater

transverse stretch travet at a greater absotute

speed

than others

if,

with gl &#x3E;

0, i)

^either g2 0 and

,f’(A2

⁺

A3) -

¹

g1/g2,

ii)

^Or g2 &#x3E; o and 1 -

(A2

⁺

A3) g1/g2 I

c)

^{The two} types

of

^wave

front

^{travel at the same absolute}

speed if

^and

only if

^the

corresponding

transverse stretches are

equal.

The result

c)

^{is exact and does not}

require

^the

approximation (5.19).

Statements

a)

^and

b)

^{follow from the} discussion of the

sign

^{of the}

right-

hand side of

Eq. (5.19).

The reason

why

^{we have} considered _gl &#x3E; 0 is made clear as follows.

Another

possibility

^for

expressing

the difference

~T2

^--

~T~

is obtained

Annales de l’Institut Henri Poincare - Section A

by reintroducing

^the

principal

^stresses

_t(2~

^and

_t(3~

^{via an}

equation

^of

the

type

^of

Eq. (5.1)

^{for those}

quantities

^{in terms of the}

eigenvalues ~2

and

~3.

^{We have}

and

Hence, Eq. (5.20)

takes the form

if

where

is Lame’s modulus and _CT is a

typical

transverse-wave

speed.

g 2’ ⁼⁼ 0 represents ^one part ^{of the} neo-Hookean

assumption (stress-strain

constitutive relation at most

explicitly

linear in

F). Equation (5.23)

^is

similar to an

equation given

^{in our}

previous

^work

[2].

It says that transverse

wave fronts with

amplitude parallel

^to the axis of lesser transverse stress travel at a greater ^absolute

speed

than the others. The other part ^{of the} neo-Hookean

assumption

^{is obtained}

by looking

^at

Eq. (5.18) which,

^with

the

approximation

^made

^above,

^{takes the same form as}

Eq. (5.23)

^{if and}

only 03C1g1/2v2,

^{where v is a}

typical

transverse stretch.

Since

^is

experi- mentally

^{shown to be}

positive (and

^must in fact be so

according

^{to the}

thermodynamics

of neo-Hookean

materials),

^and

^v2

&#x3E;

0,

^then gl must be greater ^{than zero.}

By

^{the same} token the definition

(5 .24)2

^{makes sense.}

In conclusion of

this, point

^a

representation

of neo-Hookean materials is obtained for

The statement

c)

^of

Corollary

^{5.3 holds}

good

^{in certain}

degenerate

cases of initial stresses and

strains,

^for

instance, i )

^{if this state is a}

cylindri- cally symmetric

^{one about} the direction and

ii)

if this initial state is

spherical,

^that

is, fully degenerate,

^{in which case the} above-obtained results

apply although

the notion of

principalness

has lost its

meaning.

^{Such an}

initial state

is,

^for

instance,

^{an initial state of}

high hydrostatic

pressure, ^as

can arise in the «

geophysics »

^of neutron stars

(See

^Ruderman

[7~]). Regard- ing

^this

special

^{case the}

following

remarkable result ^can be arrived at.

THEOREM 5 . 4. ^-

(simple) speed u~ and the (double) speed 0, f longitudinal

^and transverse wave

.f’ronts

^that

propagate in

^an

isotropic

relativistic nonlinear elastic

body, of

which the initial state is one

of high

Vol. XXVIII, n° 2 - 1978.

hydrostatic

pressure po

(case of

dense stellar

objects),

are related

by

^the

universal

relationship

^,

^

where

and

a

being

the sound

speed,

^and

f

the index

(in

^{the sense of}

Lichnerowicz), of

^a retativistic

perfect fluid

^that would have the same taw

of compression.

~’roof.

^{We are in a}

fully degenerate

^{case for which}

with

where ~~! ^{= C2}

^{= C. Set v =}

^{( 1 - 2~) -1 ~2}

^the

^isotropic

stretch in the state

9Ko.

^{Then the matter} proper

density

^{and the same}

density

in an

ideally

unstrained state, p~i~, ^are related

by

^the

equation

^{= 3.}

We deduce thus

and

by applying

^{the chain rule of} differentiation. It follows from

(5.30)

^and

(5.32)

^that

whereas

Eqs. (5.12)

^and

(5.16)

^{reduce to}

and

respectively,

^{on account}

of Eq. (5.31)

^{and o of the} definition

(5.28)1.

^Substi-

Anna/es de l’Institut Henri Poineare - Section A

tuting

^from

(5. 33)

^into

(5.34)

^and

combining (5. 34)

^and

(5. 35) completes

the

proof. Q.

^{E. D.}

The exact result

(5.26)

valid within the relativistic framework of finite- strain

theory

^{is universal for no}

hypotheses

need be made

concerning

^the

explicit

functional form of the internal _energy function. It consists in the

general

relativistic

generalization

^{of a} classical result due to Truesdell

[19].

In the neo-Hookean case described

by Eq. (5 . 25)

^{it reduces to the}

equation proposed by

^Carter

[7].

^It is reasonable to assume that &#x3E; 0.

Therefore,

ⁱⁿ

general, ~ ~2 ~~ &#x3E; -

Relativistic

causality

^thus

imposes

^that

(4/3)~lCl

^{+ 1. That}

^is,

It is difficult to establish the

reality

^{of but the}

following

^{can be}

pointed

out :

COROLLARY 5.5.

- If

transverse ^wave

fronts

^can propagate at ^{all in}

an

isotropic

retativistic nontinear elastic

body in an

^initial

state of hydro-

static pressure, then

longitudinal

^wave

.f’ronts

^can propagate as ^welt.

Indeed,

^if

u2|

&#x3E;

0,

^then

U2~

&#x3E;

a2

&#x3E; 0.

However,

^if

U2|

⁰

(no

propaga- tion of transverse

fronts),

^then

~~~

^{a2 and}

~~~

^{can be zero or}

^imaginary,

so that the case

~~j

⁰

^(no propagation

^of

longitudinal

^wave

fronts)

cannot a

priori

be excluded.

In the neo-Hookean case the

causality

^condition

(5.36)

^{takes on the}

simple

^form

(with

^=E

For a

body

^{unable to} support

shearing effects,

^{hence for}

,u

⁼

0,

^{this last}

inequality

^{reduces to that}

given

in relativistic

hydrodynamics (Cf.

Israel [20] ).

6. REMARK ON THE INCOMPRESSIBLE CASE

Typical

^{materials for which the}

foregoing development applies

^{are those}

which make _{up the} thick crust of neutron stars, of which the ^outer

portion probably

resembles terrestrial matter except ^{that it is about}

1018

times more

rigid

than steel and much more

incompressible,

^{so that it is easier to}

jiggle

it than to compress it

(Cf. [8], [21 ] ).

Conclusions

regarding

^this

limiting

case can be drawn

directly

from the results of

previous

sections. If the relativistic elastic

body

^is

incompressible,

^{then the} deformations it suffers

are isochoric. This is

expressed

^{in terms of the strain tensor}

by

^the

condition

Vol. XXVIII, ^n° 2 - 1978.